In this paper we derive an explicit formula for the numerical range of(non-self-adjoint) tridiagonal random operators. As a corollary we obtain thatthe numerical range of such an operator is always the convex hull of itsspectrum, this (surprisingly) holding whether or not the random operator isnormal. Furthermore, we introduce a method to compute numerical ranges of (notnecessarily random) tridiagonal operators that is based on the Schur test. In asomewhat combinatorial approach we use this method to compute the numericalrange of the square of the (generalized) Feinberg-Zee random hopping matrix toobtain an improved upper bound to the spectrum. In particular, we show that thespectrum of the Feinberg-Zee random hopping matrix is not convex.
展开▼
